Secrecy Performance Analysis of a Cognitive Network for IoT over k-μ Channels

With the development of Internet of Things (IoTs), devices are now connecting and communicating together on a heretofore unheard-of scale, forming huge heterogeneous networks of mobile IoT-enabled devices. For beyond 5G(B5G-) enabled networks, this raises concerns in terms of spectral resource allocation and associated security. Cognitive radio is one effective solution to such a spectrum sharing issue which can be adopted to these B5G networks, which works on the principle of sharing spectrum between primary and secondary users. In this paper, we develop the confidentiality of cognitive radio network (CRNs) for IoT over k-μ fading channels, with the information transmitted between secondary networks with multiple cooperative eavesdroppers, under the constraint of the maximum interference that the primary users can tolerate. All considered facilities use a single-antenna receiver. Of particular interest, the minimum limit values of secure outage probability (SOP) and the probability of strictly positive secrecy capacity (SPSC) are developed for this model in a concise form. Finally, the Monte Carlo simulations for the system are provided to support the theoretical analysis presented.


Introduction
With the recent roll-out of 5G technology globally, an everincreasing number of intelligent devices are now joining the Internet including mobile IoT devices in social, industrial, healthcare, and smart-grid nature [1]. With this rapid rise in connectivity between such devices, associated security threats and challenges are also on the increase, becoming more pressing, and need to be resolved urgently.
Traditional network encryption mechanisms can resolve security problem through various encryption algorithms at the network layer and above. However, such encryption mechanisms can no longer provide perfect security for wireless communication networks due to the complexity and time-consuming nature of the problem. Fortunately, the influence of fading channel and noise actually provides the possibility for implementing physical layer security (PLS), which has been the subject of extensive research in the literature. On the basis of [2], Wyner first proposed a model to estimate the security of communication systems [3]. For the scenarios of active eavesdropping, Ai et al. pro-vided another evaluation benchmark, namely, average secrecy capacity (ASC), over double-Rayleigh fading channels [4]. Referring to the classical Wyner eavesdropping model, SOP was given to study the security of the correlated Rician fading channels [5]. To minimize information leakage, a precoding scheme and the security of Rician fading channels were investigated by analyzing the SOP in [6]. Elsewhere, [7] studied the security capability of large-scale fading channels according to the probability of nonzero secrecy capacity (PNSC) and SOP.
The generalized fading channel can model the real transmission environment. By changing its parameters, it can represent many channel models. To account for this, a large section of the literature has studied the transmission performance and security of the generalized fading channels [8][9][10][11][12][13][14][15]. In [8], Lei et al. employed two mathematical forms to complete the derivation of the lower limit of SOP and strictly positive secrecy capacity (SPSC). Elsewhere, the ability of such a channel to resist active eavesdropping was investigated by deriving the ASC in [9]. Using a system model of decode-and-forward (DF) relay cooperation over generalized-K channels, the exact and approximate theoretical expressions of SOP and ESC were evaluated in [10]. Using channels with the premise that the main link follows α-μ distribution and the eavesdropping link was modelled as k-μ distribution, the analytical expressions of ASC, SOP, and SPSC were derived in [11]. Sun et al. described the closed form of SOP and SPSC over other k-μ shadowed fading channels in a concise form [12] and gave an approximate analysis through the method of moment matching. The authors of [13] analyzed the security of Fox's H-function fading channels by simulating the SOP and probability of nonzero secrecy capacity (PNZ). In real-world wireless communication networks (WCNs), the correlation between antennas cannot be ignored. Based on this, the security performance analysis of correlated systems over η-μ fading channels [14] and k-μ shadowed fading channels [15] has also been investigated.
More recently, nonorthogonal multiple access (NOMA) and ambient backscatter communication technology have attracted more and more attention due to the high spectral and energy efficiency for the Internet of Things. In order to investigate the reliability and security of the ambient backscatter NOMA systems considering hardware damage, the outage probability (OP) and the intercept probability (IP) were studied [16]. More practically, the ambient backscatter NOMA system under in-phase and quadrature-phase imbalance (IQI) was taken into account in [17], where the expressions for the OP and the IP are derived in closed exact analytical form [18] and the secure performance for the future beyond 5G (B5G) networks in the presence of nonlinear energy harvesters and imperfect CSI and IQI in terms of the closed form of OP and IP was studied.
Most recently, many scholars are interested in CRNs because they can make use of scarce spectrum resources without causing decoding errors to the primary user's communication. Considering a multirelay network over Nakagami-m fading channels, the authors of [19] studied the effect of three different relay schemes on the security capacity of the channel. In [20], the SOP of the single-input multipleoutput (SIMO) underlay CRNs over Rayleigh fading channels with imperfect CSI were derived and analyzed. Park et al. [21] proposed a CRN model composed of a multirelay primary network and a direct link secondary network, where the outage performance of the two networks was analyzed. The secrecy outage performance of DF-based multihop relay CRN under different parameters was investigated in the presence of imperfect CSI in [22]. The authors in [23] studied the energy distribution of CRN by analyzing spectrum sharing. Based on an underlying CRN, the derivation and analysis of SOP and SPSC are described in [24]. Combined with machine learning, a resource allocation protocol for CRN has also been proposed, and the influence of channel parameters on spectrum efficiency is presented in [25]. For conditions where the secondary network cannot interfere with the communication of the primary network, the authors in [26] took PNSC and SOP as the benchmark for studying CRN over Rayleigh fading channels. Recently, security issues are studied for popular applications such as relaying system a direct connection [27] and NOMA system [28].
As a generalized channel, k-μ fading can be equated with Rayleigh, one-sided Gaussian, Nakagami-m, and Rician fading channel [29], which can be used to simulate many wireless communication scenarios, so it is of great value to explore transmission performance. Bhargav et al. [30] analyzed the security of the wiretap system over fading channels by deriving the SOP and SPSC of the considered system. The SOP was derived based on classical Wyner's model over k-μ distribution [31]. The authors in [32] studied the statistical properties of k-μ distribution and obtained the probability density function (PDF) and cumulative distribution function (CDF) for multiple independent k-μ variables. Utilizing DF relay scheme, the SOP and SPSC in a relay system over k-μ channels were provided [33]. As an extension to [32], the authors of analyzed the secrecy outage performance of a SIMO wiretap system over k-μ channels.

Motivation and Contribution.
To date, there is negligible work presented in the literature on the CRN security assessment of multiple eavesdroppers over k-μ fading channels. Motivated by the aforementioned discussions, this paper presents such an investigation into the secrecy outage performance of CRN under multiple eavesdroppers by deriving the SOP and SPSC.
The main contributions of this paper are summarized as follows: (i) The work presents a CRN security assessment of multiple eavesdroppers over k-μ fading channels, considering multiple eavesdroppers in the cognitive radio network. It provides theoretical analysis of SOP and studies the influence of channel parameters and other parameters on the secrecy outage performance (ii) The paper also presents a derivation of SPSC for such a setup, from which it can be seen that SPSC is independent of the primary channel, and this conclusion is confirmed by simulation (iii) To further evaluate the security for the considered system, the asymptotic analysis of SOP in the high signal-to-noise ratios (SNRs) is derived in this paper. The simulation results indicate that the secrecy diversity order is equal to the main channel parameters and is not influenced by the other parameters 1.2. Organization. The rest of this paper is organized as follows. Section 2 illustrates the proposed system model. Section 3 presents the premise, including the PDFs and CDFs of the main, primary, and wiretap channel. The SOP is derived in Section 4, the asymptotic SOP is then considered in Section 5. Section 6 introduces the associated evaluation of the SPSC. In Section 7, numerical results are presented based on the Monte Carlo method to verify the theoretical analysis presented in the preceding sections. Finally, Section 8 provides the conclusion for this paper.

System Model
The system model is presented in Figure 1. It consists of one primary transmitter (PT), one primary receiver (PR), one secondary transmitter (ST), one secondary receiver (SR), and multiple eavesdroppers ðE i , i = 1, 2,⋯,L E Þ. An effective way to realize spectrum sharing is to use cognitive radio networks (CRN). There are three types of CRN: interweave, underlay, and overlay. The model in this paper adopts the underlying CRN. The system model and analysis method can also be applied to other wireless fading channels.
In this paper, we assume that the considered network functions in underlay mode, i.e., the secondary users (SUs), concurrently are entitled to use the resources of the primary network. In underlay mode, communication among the secondary networks of the CRN can be implemented, but it must be carried out under the limitation of guaranteeing the quantity of service (QoS) of the primary network. ST tries to transmit information to SR in the presence of multiple cooperative wiretappers, without reducing the communication quality of the primary network. Hence, the transmitter power P s of ST is written as where I p is the maximum interference power at PR and P max represents the peak transmit power of S restricted by designed hardware. It is assumed that there are no direct links between PT and SR and that E i ði = 1, 2,⋯,L E Þ can only eavesdrop on the signal from ST. All links of the considered system are independent, nonidentity, frequency flat, and subject to k-μ fading, with the coefficients of the channel unchanging during a transmission block. Based on these assumptions, h s,v is the channel gain from S to v, v ∈ ðd, p, e i , i ∈ ð1, 2,⋯,L E ÞÞ; the power gains can be denoted as a k-μ random variable with channel parameters ðk v , μ v Þ, assuming that all channel coefficients are integers; finally, L E is the number of wiretappers. Therefore, the received signals are where n p , n d , n ei are the additive white Gaussian noise with zero mean value and variance of σ 2 on PR, SR, and E i . From (2), the received instantaneous SNRs are For convenience, we define x = jh sp j 2 ,y d = jh sd j 2 , and y e i = From this, one can get the total wiretapped channel power gain as

Statistical Characteristics of k-μ Fading
Since all channels of the considered system experience the independent, nonidentity k-μ fading, from [26] (Equation (10)), the k-μ power probability density function (PDF) of the link from ST to SR or E i can be expressed as where z ∈ ðy d , xÞ, u ∈ ðp, dÞ, and p, and d denote the subscripts of the channel coefficient from ST to PR or SR, respectively; y d ,x is the channel power gain of the link from ST to SR or PR, respectively; I m ð⋅Þ is the modified Bessel function; and Γð⋅Þ is the Gamma function. Utilizing ( [34] Equation (8.445)), we substitute λ u = ðð1 + k u Þμ u Þ/Ω u into (6), and after some algebraic manipulations, (6) can be rewritten as According to the relation between the CDF and PDF, the CDF of the channel gain can now be derived as where γðμ u + l, λ u zÞ denotes the lower incomplete Gamma function from ([34] Equation (8.350.1)). In the considered system, all eavesdropping links, though independent, are not necessarily identical, and the cooperative eavesdroppers all apply MRC techniques, such that the total channel gain of all of the wiretap links is written as y e = ∑ L l=1 y ei , where y ei is the channel gain of the link from the transmitter to E i and L is the number of eavesdroppers. Therefore, the PDF of y e is given by [33] (Equation (3)) where a i = Ω ei /½2μ ei ð1 + k ei Þ ; χ ei = 2k ei μ ei ; U = ∑ L i=0 μ i ; L is the number of eavesdroppers; Ω ei is the average power gain of the ith wiretap link; and k ei , μ ei are the channel coefficients of the ith wiretap link. The parameters ξ and β must be carefully selected to guarantee the convergence of the series in (9). Specifically, when ξ < U/2 and β > 0, (9) will converge in any finite interval; if ξ ≥ U/2,β must be chosen as β > ð2 − U/ξÞa ðnÞ /2, to make certain the uniform convergence of (9) in any finite interval, where a ðnÞ = max fa i g ði = 1, ⋯, LÞ.

Analysis of Secrecy Outage Probability
According to information security theory, perfect secrecy connection can be guaranteed if the rate of encoding of the confidential data into code words is lower or equal to the secrecy capacity. Otherwise, the security of the information will be compromised. In this section, we focus on analyzing the SOP, which is an important performance metric of 4 Wireless Communications and Mobile Computing describing the security of the considered system; it denotes the maximum achievable rate. The secrecy capacity of CRN is where C D = log 2 ð1 + γ D Þ and C E = log 2 ð1 + γ e Þ are the instantaneous channel capacity of main and wiretap link (s), respectively. For a CRN working in underlay mode, in order to guarantee the quality of the service for the primary network, the transmitter power of ST must be constrained by the maximum interference threshold, I p , that the primary user can tolerate and its maximum transmitter power, P max , as From the definition of SOP, it can be denoted as From expression (15), we can see that the SOP is composed of two components, with reference to SOP 1 and SOP 2 . The remainder of this section will consider these 2 terms in greater detail. 4.1. SOP 1 Analysis. When P s = P max , the work mode is the same as for a normal communication system. According to probability theory, SOP 1 can be calculated as Next, we derive an expression for I 1 , from (16): where γ d = ðP s /σ 2 Þy d , γ e = ðP s /σ 2 Þy e , and P s = P max . Rearranging terms and using mathematical methods, we can obtain where Θ = e C th and α = P max /σ 2 : Taking account of the fact that I 1 > Ð ∝ 0 F D ðΘy e Þf E ðy e Þdy e , here, we derive the lower bound of I 1 as Substituting (8) and (9) into (19) and utilizing [38] (Equation (3.10.1.2)), we derived the expression of I L 1 as Applying (8) to this equation, we get From this, the lower bound of SOP 1 can be obtained by applying (20) and (21) as where Let β 2 = I p /σ 2 , after some mathematical operations similar to those employed for I 1 , we obtain the following expression for SOP 2 : where HðxÞ = Ð ∞ 0 F D ðΘy e + ððΘ − 1Þ/β 2 ÞxÞf E ðy e Þdy e . Substituting (8) into this equation, after some mathematical derivation, we can obtain HðxÞ as where Then, substituting (7) into (25), I H ðxÞ can be given by where φ = 1/ð2βλ d Θ + 1Þ; substituting (26) into (25), SOP 2 becomes From this, we can substitute (28) and (7) into I P 2 , and with the aid of binomial expansion and [34] (Equation (3.351.3)), I P 2 can be shown to be where

Analysis of Secrecy Outage Probability Asymptotic Secure Outage Probability
Although the expressions of SOP can help us perform numerical analysis on the secrecy outage performance of the considered system, asymptotic analysis can also be used to further evaluate the system performance. Therefore, we focus on the derivation of an asymptotic expression of SOP in this section and study the impact of the maximum transmit power ðP max Þ of ST and the maximum interference ðI p Þ that PU can tolerate on the secrecy communication with multiple eavesdroppers. In the high-SNR region, the asymptotic SOP can be defined as where G d = μ d denotes the secrecy diversity order and οð⋅Þ represents higher order terms. The secrecy array gain is Wireless Communications and Mobile Computing

Probability of Strictly Positive Secrecy Capacity
In information theory, the absolute security of communication can be guaranteed only when the instantaneous secrecy capacity exceeds zero. Thus, SPSC is considered to be an important indicator for measuring the secure communication system, which is given by the formula SubstitutingC th into (16), we can get where and A q and A 0 are the same as mentioned above. Then, SPSC can be obtained as From this expression of SPSC, we can see that it does not rely on the primary channel gain but is only dependent on the gain of eavesdropping channel and main channel.

Numerical Results
In this section, the curves obtained by Monte Carlo simulation of SOP for the considered system are compared with the above mathematical analysis in order to consider the impact which the different related parameters have on the security of the cognitive networks. After verification, when the value of the variable reaches 50 times, it converges to a constant value. Infinite series does not affect the simulation results. The parameters utilised in this paper are set to P max = 1 and σ = 1, and the other parameter settings are as shown in the relevant figure. Figure 2 shows the curves of the SOP for different numbers of eavesdroppers. It can be seen that the analysis results are in agreement with the simulation curves across the entire range of SNRs. In addition, the approximate curve is the tangent line of the exact theoretical results. Moreover, we can also see that the SOP will increase as the number of eavesdroppers increases. This is due to the fact that all of  Figure 3. It can be seen that SOP decreases with the increase of either k d or μ d . Moreover, we can see that with this increase in the value ofk d orμ d which means the SNR at the receiver increases, the secrecy performance will improve. Figures 5 and 6 show the influence of channel parameters ðk d , μ d Þ on SOP. In Figure 6, we plot the different curves of SOP when k d and μ d are varied. The blue curve is the difference between ðk d , μ d Þ = ð1, 1Þ and ðk d , μ d Þ = ð1, 2Þ; the red    Figure 7, it can be seen that the higher the value of I p , the better the security of the system, since ST can transmit more high power information. We can also see that there is a limitation for the SOP in the high I p . This is attributed to the fact that the maximum transmit power of STðP max Þ is equal to 1, while I p ⟶ ∞, as suggested in the above sections.
Next, without loss of generality, we plot a set of curves with different parameters to observe the effect of C th on SOP. As we can see from Figure 8, the SOP for lower C th outperforms the ones for the higher C th . This is due to the fact that SOP is the probability that secrecy capacity C s remains below the output threshold C th . The lower the threshold C th is, the smaller the corresponding probability obtained. In   Wireless Communications and Mobile Computing particular, the red curve of C th = 0:01 is almost identical to the blue curve of C th = 0:1. Figure 9 shows the SPSC for different values of Ω p . From Figure 9, we can see that the SPSC decreases with the increasing values of ðk e , μ e Þ. This can be explained by noting that the larger ðk e , μ e Þ implies a stronger signal obtained by eavesdroppers; hence, the eavesdropper SNR increases which decreases the secrecy capacity and thereby increases the SOP. In addition, we can also observe that the SPSC does not change with the variation of ðk p , μ p Þ as discussed in (38).
To sum up, the interesting conclusion can be obtained that the improvement of the confidentiality is manifested by a larger value of SPSC and a smaller value of SOP. Therefore, a smaller number of eavesdropping antennas, a larger k d , a larger μ d , and a smaller C th can improve the confidentiality of the CRN model.

Conclusions
In this paper, we have investigated the security performance for CRNs which operate in underlay mode with 5G, beyond 5G, and Internet of Things (IoT) technologies, where all channels experience independent, but not necessarily identically, k-μ fading. The exact and asymptotic theoretical expressions of SOP are derived for the considered system in the presence of multiple eavesdroppers. We also derive an equivalent expression for the SPSC of such a system. These resulting formulae show that the secrecy diversity order relies on the main channel parameter. This is corroborated with simulation results, which also prove this conclusion. Finally, Monte Carlo simulation results are presented to verify these analytical expressions and illustrate the influence which factors have on the secrecy performance versus the SNR ratio (s) of the channel (s).

Data Availability
Data used to support the findings of this study are available from the corresponding author upon request.